Method and system for tomographic imaging

ABSTRACT

Approaches are disclosed for electrical impedance tomography which apply a current to a region at two or more frequencies and acquire voltage measurements at each frequency to generate a set of multi-frequency voltage measurements. One or more images of the region are generated, using spectral constraints, based on the multi-frequency data.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a non-provisional filing of U.S. Provisional PatentApplication No. 61/649,471, entitled “METHOD OF IMAGING”, filed May 21,2012, which is herein incorporated by reference in its entirety for allpurposes.

BACKGROUND

The subject matter disclosed herein relates generally to imaging, suchas voltage or current injected electrical impedance tomography and/ormagnetic induction tomography.

Electrical impedance tomography (EIT) is a medical imaging techniqueused to non-invasively probe the internal properties of an object orsubject, such as the electrical properties of materials or internalstructures within the object or subject. For example, in EIT systems, anestimate of the distribution of electrical conductivities of probedinternal structures is made and utilized to reconstruct the conductivityand/or permittivity of the materials within the probed area or volume.In certain implementations, electrodes are used to apply an alternatingcurrent (or a voltage in other implementations) to the surface of theskin and the resulting potential is measured. Measurements are made fromdifferent points on the skin and an image of impedance within the bodyis created using image reconstruction techniques. Thus, electricalimpedance tomography provides imaging information regarding the internalelectrical properties inside a body based on voltage measurements madeat the surface of the body.

Electrical impedance tomography is a non-invasive technique for imagingphysiological and pathological body functions. The benefits of EITapplications in medicine lie in the possibility of obtaining hightemporal resolution, and in the portability and limited cost of thescanner. The main limitation is the low spatial resolution, which is dueto the reconstruction problem being generally ill-posed. The underlyingprinciple is to exploit the electrical properties of biological tissuesto extract information about the anatomy and physiology of organs. Thephysical parameter of interest in EIT is the complex impedance or realconductivity that, in the case of biological tissue, are frequencydependent. A small amount of current (or a voltage) is injected into thebody and voltage measurements are acquired using peripheral electrodes.A reconstruction algorithm is implemented to image the impedancedistribution of the subject in two or three dimensions.

BRIEF DESCRIPTION

In one embodiment, a method is provided for imaging a subject. Themethod comprises applying a current or voltage to a region of tissue attwo or more frequencies. Voltage measurements are acquired at eachfrequency to generate a set of multi-frequency voltage measurements. Oneor more images of the region of tissue are generated, using spectralconstraints, based on the multi-frequency data.

In another embodiment, a monitoring and processing system for use inelectrical impedance tomography is provided. The monitoring andprocessing system comprises a monitor configured to drive an array ofelectrodes or magnetic coils; a processor configured to receive andprocess signals from the array of electrodes or magnetic coils; and amemory configured to store one or more routines. The one or moreroutines, when executed by the processor, cause acts to be performedcomprising: driving one or more of the electrodes or magnetic coils toapply a current or voltage at two or more frequencies; acquiring voltagemeasurements at each frequency via the array of electrodes or magneticcoils to generate a set of multi-frequency voltage measurements; andgenerating, using spectral constraints, one or more images based on themulti-frequency data.

In a further embodiment, one or more non-transitory computer-readablemedia encoding routines are provided. The routines, when executed, causeacts to be performed comprising: accessing a set of multi-frequencyvoltage measurements; and generating, using spectral constraints, one ormore images based on the multi-frequency data.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the presentinvention will become better understood when the following detaileddescription is read with reference to the accompanying drawings in whichlike characters represent like parts throughout the drawings, wherein:

FIG. 1 is a schematic illustrating an embodiment of an electricalimpedance spectroscopy or tomography system, in accordance with aspectsof the present disclosure; and

FIG. 2 depicts a flow chart of an embodiment of an algorithm forgenerating electrical impedance images, in accordance with aspects ofthe present disclosure.

DETAILED DESCRIPTION

As described in more detail below, provided herein are embodiments ofelectrical impedance tomography (EIT) and/or magnetic inductiontomography (MIT) methods and systems for performing multi-frequencyimaging using spectral constraints. In one embodiment, current orvoltage is injected into or induced in the body in a wide range offrequencies, such as through a pair or multiple pairs of electrodes (inEIT) or using magnetic coils (in MIT). Voltage measurements are acquiredfor each frequency on some or all of the remaining boundary electrodesor via the magnetic coils. In one implementation, boundary voltage datais employed to directly reconstruct the distribution of each tissue,rather than the difference in conductivity between each pair offrequencies. Given that the reconstructed parameter is frequencyindependent, this approach allows for the simultaneous use ofmulti-frequency data, thus imposing more constraints for thereconstruction problem. Furthermore this approach allows for use offrequency difference data in non-linear reconstruction algorithms.

With the foregoing discussion in mind, an example of an EIT system 10 isillustrated in FIG. 1. For example, the EIT system 10 may be configuredto estimate the electrical properties (e.g., conductivity and/orpermittivity) inside a body or object using measurements obtained on thesurface of the body or object (i.e., non-invasively). Depending on thesystem, tomographic representations spatially depicting the electricalproperties (or values derived from such electrical properties) withinthe body may be generated and/or displayed.

In the depicted example, the system 10 includes a monitoring andprocessing system 11 including a monitor 12, a display 20, a processor22, and a memory 24, as well as an array of sensors (i.e., electrodes14) and communication cables 16. In the illustrated embodiment, theelectrodes 14 are provided as an array on a surface 15 of a chest of asubject 26 above an interrogation region of the subject 26 (i.e., thesubject's anatomy below the outer surface of the chest). However, itshould be noted that in other embodiments, the electrodes 14 may bepositioned on or about any desired portion of the subject's anatomy,such as but not limited to the chest, an arm, a leg, and so forth, or onany desired portion of another object or subject proximate to a desiredinterrogation region. By way of further example, in other embodimentsthe electrodes 14 may be positioned on the surface of the subject orobject, near the surface of the subject or object, or penetrating thesurface of the subject or object, depending on implementation-specificconsiderations. Accordingly, it should be noted that the electrodes 14may take on a variety of different forms, such as surface-contactingelectrodes, stand-off electrodes, capacitively coupled electrodes,conducting coils, antennas, and so forth. Additionally, the electrodesmay be arranged in any desired spatial distribution, such as lineararrays, rectangular arrays, etc.

Further, different quantities (e.g., 8, 16, 32, and so forth) andarrangements (e.g., adhesive individual electrodes, electrodes providedon a pad, etc.) of electrodes 14 may be provided in differentembodiments. In certain embodiments, coupling between the subject 26 andthe electrodes 14 may be achieved by an adhesive portion (e.g., a tackybase layer) of the electrodes 14 or a component to which the electrodes14 are attached. For example, in some embodiments, the electrodes 14 maybe provided attached to or otherwise integrated with a compliant pad orsubstrate that may be positioned or placed on the subject 26.

During operation, the electrodes 14 communicate with the monitor 12,which may include one or more driving and/or controlling circuits forcontrolling operation of the electrodes 14, such as to generateelectrical signals at each electrode. In one such embodiment, eachelectrode 14 is independently addressable by the drive or controlcircuitry. The drive and/or control functionality may be provided as oneor more application specific integrated circuits (ASICs) within themonitor 12 or may be implemented using one or more general orspecial-purpose processors 22 used to execute code or routines forperforming such control functionality.

In addition, one or more of the processors 22 may provide dataprocessing functionality with respect to the signals read out using theelectrodes 14. For example, a processor 22 may execute one or morestored processor-executable routines that may process signals derivedfrom the measured electrical signals to generate numeric values ortomographic representations for review by a user, as discussed herein.Further, the routines executed by the processor 22 and/or the dataprocessed by the processor 22 may be stored on a storage component(i.e., memory 22 or other suitable optical, magnetic, or solid-statestorage structures in communication with the processor 22). Suitablestorage structures include, but are not limited to, one or more memorychips, magnetic or solid state drives, optical disks, and so forth,suitable for short or long-term storage. The storage component may belocal or remote from the processor 22 and/or system 10. For example, thestorage component may be a memory 24 or storage device located on acomputer network that is in communication with the processing component22. In the present context, the storage component may also storeprograms and routines executed by the processing component 22, includingroutines for implementing the presently disclosed approaches.

While the foregoing generally describes aspects of an EIT imaging systemwith which the present techniques may be employed, as noted above, thepresent approaches may also be suitable for use in a magnetic inductiontomography (MIT) system. In such an MIT system, magnetic coils, asopposed to electrodes 14, may be employed to inject or induce a currentor voltage in the tissue of interest and to sense the measuredparameters. For the purpose of the present disclosure, however, othercomponents or aspects of a suitable MIT system may correspond to thoseof the generalized EIT system discussed above.

With the foregoing in mind, the following provides a brief introductionto EIT and MIT imaging principles. With respect to the mathematicalfoundation, the mathematical dependence of the conductivity distributiona of an object from the potential distribution u is described byLaplace's generalized equation:∇(σ∇u)=0  (1)Reconstructing an image of the conductivity involves solving the forwardand the inverse problems. In this context, the forward problem consistsof determining the potential from knowledge of the conductivitydistribution and the Neumann boundary conditions. The forward modelF:S_(p)→S_(d) reveals how object parameters relate to acquiredmeasurements. The objective of the inverse problem is to estimate theinternal conductivity distribution of an object from theNeumann-to-Dirichlet map. An EIT image can be obtained via least-squareminimization of a regularized objective functional φ(σ):R^(n)→R:

$\begin{matrix}{\sigma = {\begin{matrix}{argmin} \\\sigma\end{matrix}\begin{Bmatrix}\underset{︸}{{{\xi\left( {{F(\sigma)} - V} \right)}}^{2} + {{\tau\Psi}\left( {\sigma,\eta} \right)}} \\{\varphi(\sigma)}\end{Bmatrix}}} & (2)\end{matrix}$where F(σ) is the forward model, ξ ε R^(M×M) is a weighting diagonalmatrix of the residual r(σ)=F(σ)−V and Ψ(σ, η):R^(K)→R is theregularization prior.

A simple approximation of the forward problem is obtained by truncatingthe Taylor series at the first derivative and considering:F(σ)≈F(σ₀)+J(σ−σ₀)  (3)where σ₀ is a fixed baseline. The difference in boundary voltages ΔVwith respect to the baseline expressed in terms of the conductivitychange Δσ=σ−σ₀ is:ΔV=F(σ)−F(σ₀)=JΔσ.Therefore a variation in conductivity Δσ with respect to a baseline canbe reconstructed from knowledge of ΔV and the sensitivity matrix.Boundary measurements are acquired for σ and σ₀ and subtracted to obtainthe data ΔV while J is computed in σ₀. The linear approximation may beemployed to produce images of small, localized conductivity changesacross time or frequency. However errors may be introduced in the caseof large or widespread changes.

Non-linear approaches may be based on the iterative search for theglobal minimum of the objective functional. In such implementations, ateach step the forward model is run and the hypothesis for σ is updated.Methods may differ in the choice of regularization term and the criteriato select the minimization step and direction.

With respect to the various EIT imaging modalities, static imaging aimsto reconstruct absolute conductivity values from a single data set.Absolute imaging has been attempted by various groups, however highsensitivity to uncertainty in the physical model and instrumentationerrors have prevented the production of satisfactory images for use in aclinical or context. Instead, most EIT imaging is performed by referringmeasurements to a baseline and imaging conductivity changes, rather thanabsolute values. This produces contrast images, of which the absolutepixel values do not provide quantitative information. One advantage ofdifference imaging over absolute imaging is the suppression of geometricand instrumentation errors. However, a disadvantage is that a baselinemust be identified.

Dynamic and multi-frequency EIT imaging are distinguished by the choiceof baseline. In dynamic imaging, measurements are referred to a previoustime point:ΔV _(TD) =V _(t) −V _(t) ₀   (5)Time-difference EIT allows for the imaging of impedance variations overtime from small changes in the boundary voltages. In certainimplementations of clinical imaging in biomedical studies, atime-difference method has been employed and an assumption of linearitymade regarding changes in conductivity in the subject and the boundaryvoltage recording. Time-difference EIT imaging allows for monitoring ofdynamic body functions such as lung ventilation or gastric emptying.

Multi-frequency EIT, or EIT-Spectroscopy (EITS), is based on thedifferences between the impedance spectra of tissues. Measurements areacquired simultaneously or in rapid sequence by varying the frequency ofthe injected current (or voltage) and are referred to a baselinefrequency:ΔV _(FD) =V _(ω) −V _(ω) ₀ .  (6)Frequency-difference EIT allows for the imaging of an event withoutinformation regarding the condition prior to the onset. This is usefulin a medical context for producing diagnostic images of conditions suchas acute stroke or breast cancer as patients are admitted into careafter the onset of the pathology, and a baseline recording of thehealthy tissue is not available.

With the foregoing discussion of EIT imaging and systems in mind, andturning to FIG. 2, the present approach is generally directed toperforming multi-frequency EIT using spectral constraints. In oneimplementation, current or voltage is injected into the body through apair or multiple pairs of electrodes in a range of frequencies. Voltagemeasurements are acquired for each frequency on some or all of theremaining boundary electrodes. Boundary voltage data is employed todirectly reconstruct the distribution of each tissue, rather than thedifference in conductivity between each pair of frequencies. Given thatthe reconstructed parameter is frequency independent, this approachallows for the simultaneous use of multi-frequency data, thus imposingmore constraints for the reconstruction problem. Furthermore thisapproach allows for use of frequency difference data in non-linearreconstruction algorithms.

In this manner, an object may be imaged with frequency dependentconductivity distribution σ(x, t, ω), made-up of a limited number oftissues {t₁, . . . , t_(j), . . . , t_(T)} with distinct conductivityspectra. In one implementation, the object is modeled using a FiniteElement Mesh with a mesh having E elements:σ(x,t,ω)={σ₁, . . . , σ_(n), . . . , σ_(E)}.  (7)Boundary voltage measurements are acquired at multiple frequenciesω={ω₁, . . . , ω_(i), . . . , ω_(F)}.

In one embodiment, a priori knowledge of the conductivity spectra oftissues is employed to define the conductivity distribution in terms ofthe volume fraction value of tissues in the domain. If the conductivityvalues of each tissue is known exactly for each measurement frequencyσ(ω_(i), t_(j))=ε_(ij) then the conductivity of each element can beapproximated by the linear combination of the conductivity of itscomponent tissues:

$\begin{matrix}{{\sigma\left( {x,t,\omega_{i}} \right)} = {\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}\varepsilon_{ij}}}} & (8)\end{matrix}$where 0<ƒ_(nj)<1 and Σ_(j=1) ^(T)ƒ_(nj)=1.

The relationship between conductivity and boundary voltages V={V₁, . . ., V_(k), . . . , V_(D)} can be rewritten as:

$\begin{matrix}{{V\left( {\sigma,I} \right)} = {V\left( {{\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}{\varepsilon_{j}(\omega)}}},I} \right)}} & (9)\end{matrix}$therefore the chain rule yields, for j=1, . . . , T:

$\begin{matrix}{\frac{\partial{V(\omega)}}{\partial f_{j}} = {{\frac{\partial V}{\partial\sigma}\frac{\partial\sigma}{\partial f_{j}}} = {{\frac{\partial V}{\partial\sigma}\varepsilon_{j}}❘_{\omega}}}} & (10)\end{matrix}$where J(σ)={[J]_(kn)=∂V_(k)/∂σ_(n)} is the Jacobian of the forward map.

The voltage measurements are acquired simultaneously or in rapidsequence, so that the distribution of each tissue is constant:

$\begin{matrix}{{\sigma\left( {x,\omega_{i}} \right)} = {\sum\limits_{j = 1}^{T}{{f_{j}(x)}\varepsilon_{ij}}}} & (11)\end{matrix}$In certain embodiments, fraction images are reconstructed using a linearmethod if the relationship between the difference boundary voltages andchange in conductivity across frequencies is approximately linear.Otherwise a non-linear method is implemented.

With respect to the linear implementation, such an implementation issuitable for treating problems for which the assumption of linearitybetween changes in conductivity in the subject and the boundary voltagerecording is valid and the conductivity spectral of the componenttissues are known. The linear approximation of the forward map isobtained by truncating the Taylor series at the first derivative andconsidering:F(σ)≈F(σ₀)+J(σ−σ₀)  (12)where σ₀ is a fixed baseline. The difference in boundary voltages withrespect to a baseline ΔV is expressed in terms of the conductivitychange Δσ=σ−σ₀ as:ΔV(t,ω)=F(σ(x,t,ω)−F(σ₀)=J(σ₀)Δσ(x,t,ω).  (13)Therefore, based on the fraction model:

$\begin{matrix}{{\Delta\;{V\left( {t,\omega} \right)}} = {{J\left( {t_{0},\omega_{0}} \right)} \cdot {\Delta\left( {\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}{\varepsilon_{j}(\omega)}}} \right)}}} & (14)\end{matrix}$where Δ indicates a change over time or frequency. Measurements arerepeated for several frequencies ω₁, . . . , ω_(i), . . . , ω_(F) toobtain a matrix:

$\begin{matrix}{\begin{pmatrix}{V\left( \omega_{1} \right)} \\\vdots \\{V\left( \omega_{i} \right)} \\\vdots \\{V\left( \omega_{F} \right)}\end{pmatrix} = {\begin{pmatrix}{{J\left( \omega_{1} \right)}{\varepsilon_{1}\left( \omega_{1} \right)}} & \ldots & {{J\left( \omega_{1} \right)}{\varepsilon_{j}\left( \omega_{1} \right)}} & \ldots & {{J\left( \omega_{1} \right)}{\varepsilon_{T}\left( \omega_{1} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{{J\left( \omega_{i} \right)}{\varepsilon_{1}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{i} \right)}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{i} \right)}{\varepsilon_{T}\left( \omega_{1} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{{J\left( \omega_{F} \right)}{\varepsilon_{1}\left( \omega_{F} \right)}} & \ldots & {{J\left( \omega_{F} \right)}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{F} \right)}{\varepsilon_{T}\left( \omega_{F} \right)}}\end{pmatrix}\begin{pmatrix}f_{1} \\\vdots \\f_{j} \\\vdots \\f_{T}\end{pmatrix}}} & (15)\end{matrix}$

A reference frequency ω₀ is defined and the conductivity change overfrequency is imaged. Assuming that

$\frac{\partial f_{j}}{\partial t} = 0$for j=1, . . . , T then:

$\begin{matrix}\begin{matrix}{{\Delta_{\omega}\sigma} = {{{\sigma(\omega)} - {\sigma\left( \omega_{0} \right)}} =}} \\{= {{{\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}{\varepsilon_{j}(\omega)}}} - {\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}{\varepsilon_{j}\left( \omega_{0} \right)}}}} =}} \\{= {{\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}\left( {{\varepsilon_{j}(\omega)} - {\varepsilon_{j}\left( \varepsilon_{0} \right)}} \right)}} \approx}} \\{\approx {\sum\limits_{j = 1}^{T}{{f_{j}\left( {x,t} \right)}\Delta_{\omega}{\varepsilon_{j}.}}}}\end{matrix} & (16)\end{matrix}$Therefore:

$\begin{matrix}{\begin{pmatrix}f_{1} \\\vdots \\f_{j} \\\vdots \\f_{T}\end{pmatrix} = {\begin{pmatrix}{{J\left( \omega_{0_{1}} \right)}\Delta_{\omega}{\varepsilon_{1}\left( \omega_{1} \right)}} & \ldots & {{J\left( \omega_{0_{1}} \right)}\Delta_{\omega}{\varepsilon_{j}\left( \omega_{1} \right)}} & \ldots & {{J\left( \omega_{0_{1}} \right)}\Delta_{\omega}{\varepsilon_{T}\left( \omega_{0_{2}} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{{J\left( \omega_{0_{i}} \right)}\Delta_{\omega}{\varepsilon_{1}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{0_{i}} \right)}\Delta_{\omega}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{0_{i}} \right)}\Delta_{\omega}{\varepsilon_{T}\left( \omega_{1} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{{J\left( \omega_{0_{F}} \right)}\Delta_{\omega}{\varepsilon_{1}\left( \omega_{F} \right)}} & \ldots & {{J\left( \omega_{0_{F}} \right)}\Delta_{\omega}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {{J\left( \omega_{0_{F}} \right)}\Delta_{\omega}{\varepsilon_{T}\left( \omega_{F} \right)}}\end{pmatrix}^{- 1}\begin{pmatrix}{\Delta_{\omega}{V\left( \omega_{1} \right)}} \\\vdots \\{\Delta_{\omega}{V\left( \omega_{i} \right)}} \\\vdots \\{\Delta_{\omega}{V\left( \omega_{F} \right)}}\end{pmatrix}}} & (17)\end{matrix}$If the same reference conductivity σ(ω₀) is used for each frequency wehave that J=J(ω₀ ₁ )= . . . =J(ω₀ _(F) ) and equation (17) can berewritten using the Kronecker multiplication. If the matrix Δε isdefined as:

$\begin{matrix}{{{\Delta\varepsilon} = \begin{pmatrix}{\Delta_{\omega}{\varepsilon_{1}\left( \omega_{1} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{j}\left( \omega_{1} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{T}\left( \omega_{1} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{\Delta_{\omega}{\varepsilon_{1}\left( \omega_{i} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{T}\left( \omega_{i} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{\Delta_{\omega}{\varepsilon_{1}\left( \omega_{F} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{j}\left( \omega_{F} \right)}} & \ldots & {\Delta_{\omega}{\varepsilon_{T}\left( \omega_{F} \right)}}\end{pmatrix}}{{then}\text{:}}} & (18) \\{{{\Delta\varepsilon} \otimes J} = \begin{pmatrix}{J\;\Delta_{\omega}{\varepsilon_{1}\left( \omega_{1} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{j}\left( \omega_{1} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{T}\left( \omega_{1} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{J\;\Delta_{\omega}{\varepsilon_{1}\left( \omega_{i} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{j}\left( \omega_{i} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{T}\left( \omega_{i} \right)}} \\\vdots & \ddots & \vdots & \ddots & \vdots \\{J\;\Delta_{\omega}{\varepsilon_{1}\left( \omega_{F} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{j}\left( \omega_{F} \right)}} & \ldots & {J\;\Delta_{\omega}{\varepsilon_{T}\left( \omega_{F} \right)}}\end{pmatrix}} & (19)\end{matrix}$where

is the external or Kronecker multiplication. Therefore, the fractionimages f_(j)∀j=1, . . . , T are recovered using the equation:

$\begin{matrix}{{\begin{pmatrix}f_{1} \\\vdots \\f_{j} \\\vdots \\f_{T}\end{pmatrix} = {\left( {{\Delta\varepsilon} \otimes J} \right)^{- 1}\begin{pmatrix}{\Delta_{\omega}{V\left( \omega_{1} \right)}} \\\vdots \\{\Delta_{\omega}{V\left( \omega_{i} \right)}} \\\vdots \\{\Delta_{\omega}{V\left( \omega_{F} \right)}}\end{pmatrix}}}{where}} & (20) \\{\left( {{\Delta\varepsilon} \otimes J} \right)^{- 1} = {{\Delta\varepsilon}^{- 1} \otimes J^{- 1}}} & (21)\end{matrix}$is the inverse of the Kronecker product.

With respect to the non-linear implementation, the non-linearimplementation is suitable for situations where the relationship betweenchanges in conductivity in the subject and the boundary voltagerecording is non-linear and the conductivity spectral of the componenttissues are approximately known.

Fractions may be reconstructed by iteratively minimizing a regularizedobjective function using a non-linear method:

$\begin{matrix}{f = {\arg{\min\limits_{f}\left( {\varphi(f)} \right)}}} & (22)\end{matrix}$where f is the vector of fraction values ƒ_(n) _(j) , where n runs overthe elements in the mesh, and j over the tissues. The fraction modelallows for the use of frequency-difference data in the objectivefunction:φ(f)=Σ_(i)½(∥F(σ_(i))=F(σ₀)−(V _(i) −V ₀)∥²+αΨ(f))  (23)where the data can be normalized to a certain frequency. Absolute ortime-difference data can also be used.

For example using Tikhonov regularization, the objective function forfraction reconstruction becomes:

$\begin{matrix}\begin{matrix}{{\varphi(f)} = {{\frac{1}{2}{\sum\limits_{i}{{\left( {{F\left( {\sum\limits_{j}{f_{j}\varepsilon_{ij}}} \right)} - {F\left( {\sum\limits_{j}{f_{j}\varepsilon_{0j}}} \right)}} \right) - {\Delta\; V_{i}}}}^{2}}} +}} \\{{\alpha{\sum\limits_{i}{{\sum\limits_{j}\left( {f_{j}{\Delta\varepsilon}_{ij}} \right)}}^{2}}} =} \\{= {{\frac{1}{2}{\sum\limits_{ik}\left\lbrack {\left( {{F_{k}\left( {\sum\limits_{j}{f_{j}\varepsilon_{ij}}} \right)} - {F_{k}\left( {\sum\limits_{j}{f_{j}\varepsilon_{0j}}} \right)}} \right) - {\Delta\; V_{ik}}} \right\rbrack^{2}}} +}} \\{\alpha{\sum\limits_{in}\left\lbrack {\sum\limits_{j}\left( {f_{jn}{\Delta\varepsilon}_{ij}} \right)} \right\rbrack^{2}}}\end{matrix} & (24)\end{matrix}$where i runs over frequencies and k over measurements. Otherregularization methods, such as Total Variation or Markov Random Field,can be used.

In this case, the objective function is differentiable, and the gradientis:[∇f(f)]_(jn)=Σ_(ik)[(J _(kn)(Σ_(t)ƒ_(t)ε_(it))ε_(ij) −J_(kn)(Σ_(t)ƒ_(t)ε_(0j))ε_(0j))*(F _(k)(Σ_(t)ƒ_(t)ε_(ij))−F_(k)(Σ_(t)ƒ_(j)ε_(0j))−ΔV_(ik))]+αΣ_(i)[Δε_(ij)*Σ_(t)(ƒ_(tn)Δε_(it))]  (25)where J is the Jacobian of the forward map F(σ).

The Hessian is approximated by:[H(f)]_(m) _(j) _(n) _(l) =Σ_(ik)[(J _(m) _(j)_(k)(Σ_(t)ƒ_(t)ε_(it))ε_(ij) −J _(m) _(j)_(k)(Σ_(t)ƒ_(t)ε_(0t))ε_(0j))*(J _(kn) _(l) (Σ_(t)ƒ_(t)ε_(it))ε_(il) −J_(kn) _(l) (Σ_(t)ƒ_(t)ε_(0t))ε_(0l))]+αΣ_(i)(Δε_(ij)*Δε_(il))  (26)where m_(j) and n_(l) run over all elements and tissues. Therefore thefraction images f_(j)∀_(j)=1, . . . , T may be reconstructedsimultaneously by using a second order descent method, such as DampedGauss-Newton or Non-linear Conjugate Gradients. or Gradient Projection.The reconstruction can be constrained so that 0<ƒ_(nj)<1 and Σ_(j=1)^(T)ƒ_(nj)=1.

With the foregoing in mind, and turning to FIG. 2, the presentdisclosure provides a method 48 of imaging a subject that suitable foruse in electrical impedance tomography. In one implementation, thesubject is modeled using a finite element approach. In certainembodiments, the region of the subject (e.g., patient) being imagedconsists of a plurality of tissue types (i.e., two or more types oftissue). In such embodiments, to the extent that predetermined tissueconductivity spectra are available and are used in the imagingoperations associated with the method 48, such predetermined tissueconductivity spectra consist of known or predetermined spectralconductivity data for at least the tissue types expected to be presentin the subject within the imaged region. In one implementation, currentor voltage is injected (block 50) into the tissues of the subject, suchas through a pair or multiple pairs of electrodes, in a range offrequencies.

Voltage measurements 60 are acquired (block 58) for each frequency onsome or all of the remaining boundary electrodes. In one embodiment, theboundary voltage measurements 60 comprise a set of multi-frequencymeasurements. In certain implementations, the boundary voltagemeasurements 60 of the subject are obtained at a single measurement timerather than at two or more measurement times separated by some intervalof time. Such an implementation may offer advantages since a singlemulti-frequency measurement of the subject can be used to construct animage, rather than taking two or more measurement separated in time.

Turning back to FIG. 2, a set of frequency-difference data 74 iscalculated (block 70) from the set of multi-frequency measurements. Inone implementation, the step of calculating (block 70)frequency-difference data includes identifying a baseline from the setof multi-frequency measurements and calculating the frequency-differencedata with respect to the baseline.

The frequency-difference data 74 is used to generate (block 80) one ormore images 82 of the subject. In one embodiment, the act of generating(block 80) the images 82 includes the step of forming an image 82 usinga frequency independent approach that employees the frequency-differencedata, as discussed herein. In addition, in certain implementations, theact of generating the images 82 may employ spectral constraints, asdiscussed herein. The act of generating images 82 may include the act ofdefining the conductivity distribution of the subject in terms of avolume fraction of tissues, as discussed above. Similarly, the act ofgenerating images 82 may use the boundary voltage measurements 60 toreconstruct the volume fraction distribution of each tissue or tissuetype within the region undergoing imaging, as discussed herein.

The disclosed method may be useful for generating diagnostic images ofconditions such as acute stroke or breast cancer where prior images ordata may be unavailable. That is, in general, this approach may beuseful for patients that are admitted into care after the onset of apathology, where a previous recording of healthy tissue is notavailable. As the present method uses frequency differences to constructan image rather than measurements taken at different times, imaging canbe performed quickly and absolute conductivity values can be gained froma single data set. In certain implementations, the boundary voltage datamay be used to directly reconstruct the volume fraction distribution ofeach component tissue, rather than the difference in conductivitybetween pairs of frequencies.

Technical effects of the invention include performing multi-frequencyEIT using spectral constraints. Voltage measurements are acquired for arange of frequencies using electrodes disposed at various points on thepatient. The boundary voltage data is used to directly reconstruct thedistribution of each tissue, rather than the difference in conductivitybetween each pair of frequencies. This approach allows for thesimultaneous use of multi-frequency data, thus imposing more constraintsfor the reconstruction problem.

This written description uses examples to disclose the invention,including the best mode, and also to enable any person skilled in theart to practice the invention, including making and using any devices orsystems and performing any incorporated methods. The patentable scope ofthe invention is defined by the claims, and may include other examplesthat occur to those skilled in the art. Such other examples are intendedto be within the scope of the claims if they have structural elementsthat do not differ from the literal language of the claims, or if theyinclude equivalent structural elements with insubstantial differencesfrom the literal languages of the claims.

The invention claimed is:
 1. A method for imaging a subject, comprising:applying a current or voltage to a region of tissue comprising multipletissue types at two or more frequencies; acquiring voltage measurementsat each frequency of a set of frequencies to generate a set ofmulti-frequency voltage measurements; and generating, using spectralconstraints, one image of the region of tissue based on simultaneous useof the set of multi-frequency voltage measurements, wherein the spectralconstraints are based on prior knowledge of the differences in theimpedance of the tissue types of the region of tissue with respect tothe differences in the frequencies of the set of frequencies.
 2. Themethod of claim 1, comprising modeling the region of tissue using afinite element mesh and applying a volume fraction describing adistribution of the tissue types to each element of the finite elementmesh to estimate a conductivity distribution.
 3. The method of claim 1,wherein the current is applied to the region of tissue and the voltagemeasurements acquired via a pair or multiple pairs of electrodes.
 4. Themethod of claim 1, wherein the voltage measurements at each frequencyare acquired simultaneously.
 5. The method of claim 1, wherein thevoltage measurements at each frequency are acquired at two or moremeasurement times.
 6. The method of claim 1, further comprising:calculating frequency-difference data from the set of multi-frequencyvoltage measurements; wherein generating the one image of the region ofthe tissue is based on the frequency difference data.
 7. The method ofclaim 6, wherein calculating frequency difference data comprisesidentifying a baseline from the set of multi-frequency voltagemeasurements and calculating the frequency-difference data with respectto the baseline.
 8. The method of claim 1, wherein generating the oneimage of the region of the tissue comprises defining a conductivitydistribution of the region in terms of a volume fraction of the tissuetypes.
 9. The method of claim 1, wherein generating the one image of theregion of the tissue comprises using the multi-frequency voltagemeasurements to reconstruct a volume fraction distribution of eachtissue type within the region of the tissue.
 10. A monitoring andprocessing system for use in tomography, comprising: a monitorconfigured to drive an array of electrodes or magnetic coils; aprocessor configured to receive and process signals from the array ofelectrodes or magnetic coils; and a memory configured to store one ormore routines which, when executed by the processor, cause acts to beperformed comprising: driving one or more of the electrodes or magneticcoils to apply a current or voltage at two or more frequencies;acquiring voltage measurements at each frequency via the array ofelectrodes or magnetic coils to generate a set of multi-frequencyvoltage measurements; and generating, using spectral constraints, oneimage of a region of tissue based on simultaneous use of themulti-frequency voltage measurements, wherein the spectral constraintsare based on prior knowledge of the differences in the impedance for oftissue types of the region of tissue with respect to differences betweenthe two or more frequencies.
 11. The monitoring and processing system ofclaim 10, wherein the one or more routines, when executed by theprocessor, cause further acts to be performed comprising displaying theone or more images on a display of the monitor.
 12. The monitoring andprocessing system of claim 10, wherein the voltage measurements at eachfrequency are acquired simultaneously.
 13. The monitoring and processingsystem of claim 10, wherein the one or more routines, when executed bythe processor, cause further acts to be performed comprising:calculating frequency-difference data from the set of multi-frequencyvoltage measurements; wherein generating the image of a region of tissueis based on the frequency difference data.
 14. The monitoring andprocessing system of claim 13, wherein calculating frequency differencedata comprises identifying a baseline from the set of multi-frequencyvoltage measurements and calculating the frequency-difference data withrespect to the baseline.
 15. The monitoring and processing system ofclaim 10, wherein generating the one image comprises using themulti-frequency voltage measurements to reconstruct a volume fractiondistribution of each tissue type within the region.
 16. The monitoringand processing system of claim 10, wherein generating the one imagecomprises defining a conductivity distribution of the region in terms ofa volume fraction of tissue types.
 17. One or more non-transitorycomputer-readable media encoding routines which, when executed, causeacts to be performed comprising: accessing a set of multi-frequencyvoltage measurements obtained at multiple frequencies of a set offrequencies; and generating, using spectral constraints, one image basedon simultaneous use of the multi-frequency data, wherein the spectralconstraints are based on prior knowledge of the differences in impedanceof the tissue types of the region of tissue with respect to thedifferences between the frequencies of the set of frequencies.
 18. Theone or more non-transitory computer-readable media of claim 17, whereinthe routines, when executed, cause further acts to be performedcomprising: driving one or more of the electrodes to apply a current attwo or more frequencies of the set of frequencies; and acquiring voltagemeasurements at each frequency via the array of electrodes to generatethe set of multi-frequency voltage measurements.
 19. The one or morenon-transitory computer-readable media of claim 17, wherein the set ofmulti-frequency voltage measurements are acquired simultaneously. 20.The one or more non-transitory computer-readable media of claim 17,wherein the routines, when executed, cause further acts to be performedcomprising: calculating frequency-difference data from the set ofmulti-frequency voltage measurements; wherein generating the one imageis based on the frequency difference data.
 21. The one or morenon-transitory computer-readable media of claim 20, wherein calculatingfrequency difference data comprises identifying a baseline from the setof multi-frequency voltage measurements and calculating thefrequency-difference data with respect to the baseline.